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Dear all,

I use R (2.15.3) with Mike Cheung’s metaSEM package (0.8-4) and thus, OpenMx (1.3.2-2301). I apply Mike Cheung’s tssem1 and tssem2 commands to perform random effects MASEM. My data set is very similar to Becker and Schram (1994), so I will use the Becker and Schram (1994) data set as it is displayed on Mike Cheung’s website (data set) and make one minor adjustment to this data set: I assume that there was an additional variable, namely class size (SIZE). I also assume that the correlation coefficients between class size and the three other variables were available in the original 10 data sets which Becker and Schram (1994) used. This data set would look like this, for example (all class size data are made up):

Becker94test <- structure(list(data = structure(list( `Becker (1978) Females` = structure(c(1, 0.47, -0.21, 0.12, 0.47, 1, -0.15, -0.05, -0.21, -0.15, 1, 0.47, 0.12, -0.05, 0.47, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Becker (1978) Males` = structure(c(1, 0.28, 0.19, 0.23, 0.28, 1, 0.18, 0.3, 0.19, 0.18, 1, 0.18, 0.23, 0.3, 0.18, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Berry (1957) Females` = structure(c(1, 0.48, 0.41, -0.1, 0.48, 1, 0.26, 0.17, 0.41, 0.26, 1, 0.34, -0.1, 0.17, 0.34, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Berry (1957) Males` = structure(c(1, 0.37, 0.4, 0.05, 0.37, 1, 0.27, 0.45, 0.4, 0.27, 1, -0.23, 0.05, 0.45, -0.23, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Rosenberg (1981) Females` = structure(c(1, 0.42, 0.48, 0.34, 0.42, 1, 0.23, 0.23, 0.48, 0.23, 1, 0.31, 0.34, 0.23, 0.31, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Rosenberg (1981) Males` = structure(c(1, 0.41, 0.74, 0.27, 0.41, 1, 0.44, 0.14, 0.74, 0.44, 1, 0.56, 0.27, 0.14, 0.56, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Weiner A (1984) Females` = structure(c(1, 0.26, 0.72, -0.03, 0.26, 1, 0.36, 0.35, 0.72, 0.36, 1, 0.24, -0.03, 0.35, 0.24, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Weiner A (1984) Males` = structure(c(1, 0.32, 0.52, 0.56, 0.32, 1, 0.1, 0.22, 0.52, 0.1, 1, 0.08, 0.56, 0.22, 0.08, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Weiner B (1984) Females` = structure(c(1, 0.58, 0.64, 0.08, 0.58, 1, 0.4, 0.63, 0.64, 0.4, 1, 0.6, 0.08, 0.63, 0.6, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"))), `Weiner B (1984) Males` = structure(c(1, 0.34, 0.28, 0.42, 0.34, 1, -0.03, 0.46, 0.28, -0.03, 1, 0.21, 0.42, 0.46, 0.21, 1), .Dim = c(4L, 4L), .Dimnames = list(c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE"), c("SAT (Math)", "Spatial", "SAT (Verbal)", "SIZE")))), .Names = c("Becker (1978) Females", "Becker (1978) Males", "Berry (1957) Females", "Berry (1957) Males", "Rosenberg (1981) Females", "Rosenberg (1981) Males", "Weiner A (1984) Females", "Weiner A (1984) Males", "Weiner B (1984) Females", "Weiner B (1984) Males")), n = c(74, 153, 48, 55, 51, 18, 27, 43, 35, 34), gender = c("Females", "Males", "Females", "Males", "Females", "Males", "Females", "Males", "Females", "Males")), .Names = c("data", "n", "gender"))

The model (based on Cheung, 2013) would look like this (see attachment if not displayed here):

I want to include SIZE as a control variable and treat the variable as follows. Apart from SIZE, everything is similar to (code).

## Stage 1 analysis: A diagonal matrix for random effects random1 <- tssem1(Becker94test$data, Becker94test$n, method="REM", RE.type="Diag") summary(random1) coef(random1) ## Prepare models for stage 2 analysis # A1: asymmetric matrix (regression coefficients) A1 <- create.mxMatrix(c(0,0,0,0, "0.2*Spatial2Math",0,0,0, "0.2*Verbal2Math",0,0,0, "0.2*SIZE2Math",0,0,0), type="Full", ncol=4, nrow=4, name="A1") A1 # S1: symmetric matrix (variance covariance matrix among variables) S1 <- create.mxMatrix(c("0.2*ErrorVarMath",0,0,0, 1,"0.2*CorBetweenSpatialVerbal",0, 1,0, 1), type="Symm", name="S1") S1 ## Stage 2 analysis (A1, S1) random2 <- tssem2(random1, Amatrix=A1, Smatrix=S1, diag.constraints=TRUE, intervals="LB") summary(random2)

Results (please do not concentrate on the values as I made up the SIZE data for explanatory reasons):

> summary(random2) Call: wls(Cov = pooledS, asyCov = asyCov, n = tssem1.obj$total.n, Amatrix = Amatrix, Smatrix = Smatrix, Fmatrix = Fmatrix, diag.constraints = diag.constraints, cor.analysis = cor.analysis, intervals.type = intervals.type, mx.algebras = mx.algebras, model.name = model.name, suppressWarnings = suppressWarnings) 95% confidence intervals: Likelihood-based statistic Coefficients: Estimate Std.Error lbound ubound z value Pr(>|z|) Amatrix[1,2] 0.298619 NA 0.217628 0.377193 NA NA Amatrix[1,3] 0.369036 NA 0.207887 0.529343 NA NA Amatrix[1,4] 0.117999 NA -0.032000 0.267997 NA NA Smatrix[1,1] 0.721916 NA 0.568998 0.831465 NA NA Smatrix[2,3] 0.176044 NA 0.080572 0.271520 NA NA Goodness-of-fit indices: Value Sample size 538.0000 Chi-square of target model 35.1684 DF of target model 2.0000 p value of target model 0.0000 Number of constraints imposed on "Smatrix" 1.0000 DF manually adjusted 0.0000 Chi-square of independence model 152.0658 DF of independence model 6.0000 RMSEA 0.1757 SRMR 0.1796 TLI 0.3188 CFI 0.7729 AIC 31.1684 BIC 22.5927 OpenMx status1: 0 ("0" and "1": considered fine; other values indicate problems)

Because I assume that SIZE was used as a control variable in the original studies, I interpret the results of Amatrix[1,4] as regression coefficient from SIZE to Math (similarly to Cheung (2013), Table 2). Am I right to treat SIZE in a similar way as I treat the independent variables Spatial and Verbal for this purpose, although SIZE is a control variable?

Thanks for your help in advance!

Regards, Eberhard

References

Becker BJ, Schram CM. 1994. Examining explanatory models through research synthesis. In The handbook of research synthesis. Cooper H, Hedges L (eds.), Sage: New York, NY.

Cheung MW-L. 2013. Fixed- and random-effects meta-analytic structural equation modeling: Examples and analyses in R. Behavior Research Methods: 1-12.

There is not much difference between control variables and independent variables. Your interpretations look correct.

However, S1 was specified incorrectly. The df of the "regression model" in summary(random2) is 2. It is because the correlation between SIZE and spatial and between SIZE and verable are fixed at 0.

The independent variables are usually correlated. Attached is the correct S1 for your reference.

Mike, thanks again for your quick answer which totally makes sense to me! Regards, Eberhard